J. Piaget's

Genetic Epistemology

And the Teaching of

Elementary Mathematics

As Analyzed by Brett W. Baskovich

Abstract: Generally, science is the analysis and understanding of the way certain events occur and can be controlled. Epistemology examines our ability to make these analyses, deducing the nature, limits and flaws of human reasoning. The science of genetic epistemology studies the advancement of this ability throughout a person's life, for our knowledge and reasoning abilities are dynamic, and when their workings are properly understood, they are more easily improved.

I. Introduction

It is common knowledge that intelligence increases with age. Additional experiences and extended cognitive development lead to more sophisticated and accurate analyses of the world [6]. But the mechanics and extent of this change are not common knowledge. And before the studies of Jean Piaget (1896-1980), they were hardly even considered. Of course, this development occurs whatever the environment of a person, but the speed and extent of development can be maximized through application of these studies [4].

In his books, Piaget defines several stages through which a child passes, each new stage representing an improvement in reasoning. He provides examples of experiments and questions applied to different age groups that clearly display the changes, and describes the implications. Literature and new teaching methods have resulted from his works. What follows shall be a brief and general description of the stages important to psychological and mathematical development and some of the consequences of this knowledge.

II. The Five Periods of Development

[This section is based primarily on Ruth Beard's An Outline of Piaget's Developmental Psychology for Students and Teachers.]

The first two years of a child's life are designated (1) the sensori-motor period. This begins with simple repeated ôreflex exercisesö, such as grasping. After about a month (these time periods can vary with environment), the infant shows his* first acquired adaptations, and can repeat and reverse successful actions (ôprimary circular reactionsö) in well-defined sequences, ôschemasö. In half a year the infant begins to understand causality; he can interact with his environment to bring about a desired change (ôsecondary circular reactionsö). While at first the change is immediate, soon after he can work towards and expect a delayed result. At this time, he can also associate an object with its spatial position, looking for something where it was last seen. At about two years, he begins to experiment, repeating actions with slight variations to test the different effects (ôtertiary circular reactionsö). Problems are more easily solved, space and causation are better understood, and memory is improved. Finally, he obtains the ability to evoke schemas mentally, facing problems with immediate solutions.

The sensori-motor period is followed by (2) the pre-conceptual substage (two to four years of age), when visual signals are most important to the child, motor connections are still significant, and speech is inconsistent and not entirely understood, though it will improve most quickly through interaction with adults. At this time, actions forced by an adult (i.e. without an explanation or known purpose) can be permanently restrictive, and it is important that the child understand why he must do what he is told to. Rational thought makes little appearance; he links together unrelated events (syncretism), attributes all things to human cause (artificialism), sees all objects as possessing life, and does not understand point-of-view. As aids during this stage, objects of various sizes are important for space, construction, and counting, as well as water, sand, and bricks (or objects with similar properties); and paint for the advancement of representation, which is later important to internalization, representing objects with other objects and symbols, the basis for algebra.

In (3) the intuitive substage (four to seven years of age), the child begins to make comparisons, but fails to see the differences in properties of an object based on perspective and lacks direction in speech. Successive opinions can differ and contradict each other, and cause and effect are often confused. He cannot make relations, such as that between a whole and its parts, and lacks ability to make one-one correspondence and measurements. Conventions, such as rules of a game, are accepted as immutable, and morals by fear of punishment. Haptic perception (identification by touch) improves, and the same aids used in the previous stage are still useful.

(4) The sub-period of concrete operations (at about eleven years) follows, when physical actions are successfully internalized as mental operations. The child can now make and understand classifications, groupings, and substitutions, but cannot use two axes of reference in representations, necessary to understand area and volume. Verbal problems are difficult, and he cannot see general laws or properly express a definition.

Finally, in (5) the period of formal operations (at about fourteen years), the adolescent can make systematic analyses, making a proposition and surveying the possibilities. He can make generalizations and laws, understand infinity, and see relations among relations. Conventions and rules are understood for what they are and can be opposed. Reasoning skills are still incomplete, and there may be flaws with aspects such as volume, but the most important development is now complete.

Such are the major periods of development outlined by Piaget. Of course, the transitions are not distinct or brief, but this categorization can be useful in understanding the changes and providing mental stimulation at each level.

* Male pronouns are used only for rhetorical convenience; Piaget's tests were performed for each gender with similar results.

III. Conception of Number

[This section is based primarily on Jean Piaget's Conception of Number.]

Towards the understanding of mathematical development, Piaget conducted various experiments in which he provided tactual representations of mathematical concepts such as conservation of quantities, one-one correspondence, ordination, cardination, classification, and additive and multiplicative composition of numbers. The benefit of these tests over simply asking a child the solution of a mathematical expression, such as the product of seven and eight, is that memorization, and even a general understanding of what product means (eight added seven times), does not necessarily imply any real comprehension of what this operation symbolizes. Piaget is testing not the child's ability to solve math problems, but his ability to solve spatial relationships that are a precursor to true understanding of mathematical operations.

Most important to any type of numerical comprehension is the concept of conservation of quantities. Piaget tested this for both continuous and discontinuous quantities.

Conservation of Continuous Quantities

(Picture from Piaget's Theory of Cognitive Development)

To test for conservation of continuous quantities, a liquid of the same amount exists in two identical containers, and the child acknowledges that the amounts are equal. Then, the liquid in one is poured into a container of different shape or multiple containers, and the child is asked which container has more liquid. Piaget places the children into three stages based on their responses.

The youngest children, in the first stage, generally identify the taller container or the more numerous containers as having more liquid. If the transfer is to one container, they can only reason in one dimension- in this case, height- and this perception is more important than whatever concepts of conservation they might have. Likewise, the presence of more containers is assumed to indicate more liquid. Children in the second stage (about four years old) may recognize conservation if height differences are small or if the original liquid is poured into only two smaller containers, but more extreme changes lead to failure. The third stage (about six years old) signifies an understanding of conservation; children can see that an increase in height can be compensated for by a decrease in width, and multiple containers do not change the volume of the liquid.

This previous study could be considered inconclusive, as continuous quantities do not precisely resemble numbers, and perhaps a child would actually think that the liquid expanded. Fortunately, it was repeated with discontinuous quantities- namely, beads. It is found that the stages correspond to those with continuous quantities. The child and observer put an equal number of beads one at a time into two containers so that they certainly match in number, but when the container is changed, children in the first stage have the same problems, believing the number to increase, and those in the second stage will contemplate the former equivalence but still decide for inequality.

A similar experiment, designed to more specifically deal with one-one correspondence, involved two rows of equal numbers of glasses.

One-One Correspondence

(Picture from Piaget's Theory of Cognitive Development)

The glasses are moved apart, and the child is asked which row has more glasses. Children in the first stage respond by indicating the longer one. They even believe they can make the longer row have fewer glasses by compacting it. Children in the second stage may see equivalent numbers if the glasses are matched oppositely, but will change their minds spontaneously. Those in the third make the correspondence, realizing that the positions of the glasses do not affect the number.

Without understanding of conservation and one-one correspondence, there can be no understanding of mathematics. A three-year-old is perfectly capable of counting, but if he sees a number as freely changing and cannot even make the most obvious equivalence (i.e. 10=10), he can do no more than memorize facts that may be of use later.

Thus far, it should be clear that to a young child, an object's shape is more important than any numerical properties. Piaget examined this specifically through the use of objects with obvious cardinal components; each is made of a clear number of counters (pieces), and all are of different shapes. The child is given one, and asked to choose the same number of counters from the collection of objects. Naturally, children in the first stage make a global comparison of the figures, attempting to make something that looks similar from the objects available. Those in the second stage make intuitive correspondence, carefully arranging the new objects so that they form the same figure as the original object; the result may be accurate, but if one is so much as turned and the shapes are no longer the same they are no longer sure of the equivalence. In the third stage, the cardinal correspondence is entirely operational; the child can select new objects based on number of counters alone. This affirms the earlier concept that in the first stages, number is not known, only approximations of perceptual wholes.

Another primary use of number is in seriation and ordination; objects are arranged in a series, and each is assigned a number. To test understanding of this, the child is provided with dolls and sticks of various sizes; the child is made to create a serial correspondence between dolls and sticks by arranging each set in order of increasing size. The child is asked to find the stick corresponding to a doll while the corresponding series are in order (testing serial correspondence), reversed, and disarranged (testing ordinal correspondence). There is the usual progression; children in the first stage have problems arranging the series, and can only select the correct stick when the series are in order. Children in the second stage can arrange the series successfully, but if it is reversed they make systematic errors in counting to find each doll's stick. In the third stage, the correspondence becomes obvious: the inverse ordinal correspondence when the order is reversed, and the cardinal correspondence when the series is randomized. Thus, by the experiment, it is seen that serial correspondence precedes ordinal and cardinal correspondence, which coincide.

For number to be fully realized, however, ordination and cardination must be coordinated and applied simultaneously, when a number intrinsic to an object denotes both a property and a position; this is most obviously used in mathematics as the equation of a line, where each number in a sequence has another corresponding value that can be derived from it; of course, Piaget's tests were more direct. In an experiment, the child is told to place ten sticks in order of increasing height to make a staircase; a doll is then made to ôclimbö the staircase, and the child is asked how many sticks were climbed and how many left. In the first stage, children cannot even properly arrange the sticks; often, they will arrange the top of the row to increase but neglect the bottom; this implies that they not only reason in one dimension (as shown in earlier tests), but that they cannot create a point of reference and so even this is incomplete. When asked for the number of sticks climbed, they make an arbitrary estimate. In the second stage, the child can order the sticks by trial and error, but will have problems inserting a new one, as this requires several simultaneous relations. Number is still not understood; the series is made intuitively by ôa juxtaposition of perceptual relationsö. The child can accurately state the number climbed with the series intact by counting. In the third stage, the series is easily constructed, and the fact that when the doll is on the nth step, n-1 steps were climbed, becomes obvious; even with the series disarranged, the former position in the series (ordinal) can be inferred from the size (cardinal) without reconstruction of the series.

In a similar experiment, there is a row of cards, where the length of the initial card is m, and the length of a card at position n is n*m. This relation is observed by all children, yet those in the first stage cannot answer a question of how many cards of length m could be made with card n without counting, and those in the second stage can only make an immediate answer if asked in increasing order. The coordination between cardination and ordination appears fully in the third stage.

The next topic addressed by Piaget was the additive composition of classes. The child is provided with a class (B), i.e. wooden beads, and two subclasses (A and A', where A is much greater in number than A'), brown beads and white beads. The child is asked which set would make the longer necklace- wooden beads or brown beads. In the first stage, the answer is always ôbrown beadsö, the sub-class A; he fails to see that A + A' = B, because as soon as he compares the parts, A and A', he loses perspective of the whole, B, and sees it as the opposite of the sub-class in question. To further investigate this, the child is asked to draw the necklaces that could be made from the wooden beads and from the brown beads; both are drawn correctly, yet when he is asked which necklace would be longer, he still responds with ôthe brown oneö; he says that the wooden necklace would be shorter because all of the brown beads are taken up to make the brown necklace.

Much of human understanding is due to the ability to simulate events and concepts mentally, returning to the initial conditions and contemplating the possible outcomes. In this first stage, the child can, in his mind, simulate the creation of one necklace; yet he cannot return to the initial conditions, making the brown beads once again available to construct the wooden necklace, for in his mind they are used up; he lacks reversibility of this operation.

In the second stage, the child may answer inaccurately, but with questions by the observer he can be made to see that the class of wooden beads includes the brown beads. In the third stage, understanding is complete.

Addition requires the understanding that a number is composed of other numbers (similar to sub-classes composing a class). To study this, the child is told he is given four sweets in the morning and four at night, then one in the morning and seven at night the next day, and asked if he receives the same number each day (the sweets are placed in front of him to visualize the situation). A child in the first stage will say he receives more the second day, as the most obvious relation is that 7>4; however, if the observer points out that 1<4, he will change his opinion to the opposite. A child in the second stage can rearrange the sweets to see that the numbers are equivalent, and a child in the third stage will see the additive properties immediately.

For further analysis, the child is given two unequal sets of counters and is told to make them equal. In the first stage, the child will attempt to move the counters between sets until they are equal; the problem arises in the child's failure to understand that when a counter is moved, it is both subtracted from one and added to the other, so he will move two counters if one set has two more than the other, and will repeat this. In the second stage, the correct result is achieved more quickly, but by geometrical arrangement. In the third stage, the child can count the difference and immediately make the correct transfer.

Piaget's next topic is that of the coordination of relations, or syllogistic logic; the concept that if X = Y, and Y = Z, then X = Z. For the experiment, there are ten vases, ten blue flowers, and ten pink flowers. A blue flower is placed in each vase, then all are taken out; this is done for the pink flowers separately, and the child is asked if there are the same number of blue and pink flowers, or, if one of each can be placed in each vase. Predictably, children in the first stage simply don't know; those in the second say there are the same because they ôall go inö; those in the third stage immediately declare that there are ten of everything after only counting one set.

To more directly apply this to multiplication, the child is told that each flower will be placed in a tube and the tubes into the vases; he is to choose an appropriate number of tubes and place them in the vases. A child in the first stage will choose a number of tubes between ten and twenty (there are still twenty flowers), realizing there are not enough when placing the flowers in. When he is to put the twenty flowers into the ten vases, he begins putting one in each, then two, then three, eventually having two in each vase by trial and error. In the second stage, the child selects ten tubes at first, but progresses immediately to twenty; likewise, when placing the flowers in the vases, he begins putting one in each vase but soon realizes that there must be two. In the third stage, the two-to-one ratio is immediately understood, as well as that for greater numbers, such as that for forty flowers, four would be put into each of ten vases. The concept of multiplication is then fully realized.

The final tests, of the additive and multiplicative composition of relations, bear resemblance to the first tests of conservation, but are slightly more complex and are now analyzed in a different context. The child is provided with two or three differing containers with equal volumes of liquid and two identical containers, and is asked to rank the filled containers by volume of liquid. In the first stage, children will make a guess at which has more; when directed to pour the liquids into the empty, identical containers, they see the equivalence, but once the liquids are poured back, they return to their original claim. The responses of children in the second stage are similar, but when the liquid is returned from the identical containers to the original ones, they become confused by the seemingly contradictory perceptions. Children in the third stage usually make an inaccurate first guess, but are sure of the equivalence once it is tested; if they are given three filled containers to start with, they need only make two comparisons to reach their conclusions.

In the final test to be described here, a container (of which there are two) is used to fill a container of twice the size. In the first stage, the child will intuitively say that the large container and two smaller containers have the same amount of liquid, but will then decide that the smaller containers have more. In the second stage, with much direction and contemplation, the child will see that the two smaller containers equal the larger containers, and in the third stage the response is immediate.

The ideas of conservation and multiplication of relations coincide; to see that when a liquid's container changes, its volume remains constant, there must be the notion that the change in one dimension is compensated by an inverse change in another. As stated by Piaget, measurement is merely the reduction of multiplications to the equalization of differences.

IV. Implications of Piaget's Theory

Piaget realized that the time periods of the stages vary by child and environment; thus, the environment should be set up in a way so as to maximize the speed of advancement and understanding [2]. The techniques that follow were suggested by Piaget, specifically addressing the field of elementary mathematics.

It is of great importance that unfamiliar ideas are not imposed on a child. He should be guided into finding the relations being studied; the ideas and processes must be understood completely before he is introduced to the formal notation and complexities. Incompletely developed operations should not be overlooked and left to use only by repetition [4].

Deductive reasoning should follow from experience with the mathematical entities, and from this the understanding of mathematics. Methods of reasoning, not necessarily specific concepts, should be taught; the concepts are best left to be found by the child through investigation, while the instructor provides direction [4].

The children should be trained in self-checking, approximation, and primarily reflection and reasoning. The instructor should study their mistakes to better understand their thought [5].

Intellect is seen to develop through interaction with environment; thus, interaction which stimulates mathematical thought should be encouraged [5]. Visual and tactual representations of the concepts- such as those used by Piaget in his studies- will aid in comprehension. The children should perform real actions with the learning materials, such as counting and exchanging pennies [3]. As most of the mathematical advancement addressed here occurs in the intuitive substage (3), spatial relations are important; bricks, water, sand, and other items of construction should be provided [2].

Naturally, most of the Piagetian methods described in literature are not by Piaget, though most follow his teachings closely. The ideas that follow are based primarily on Constance Kamii's Number in Preschool and Kindergarten: Educational Implications of Piaget's Theory.

A child should never be ordered to count or told precisely what to do to solve a problem. He should merely be encouraged to quantify objects, and ôlogico-mathematical knowledgeö will develop. The ability to use symbols, such as | | | | | | | | for eight, and signs, such as 8 for eight, coincides with this knowledge. As the use of signs is conventional, it is not as important as the natural use of symbols.

The child should be encouraged to put objects, events, and actions into relationships, and to think actively about everyday living with minimal interference. Children should be allowed to resolve conflicts by themselves; i.e. if two children are fighting over a toy, the instructor should not decide what is to be done, but should move it out of reach and tell the children to decide what is to be done with it before it is given back. Actions such as this encourage the ability to reason over the ability to follow instruction.

The numbers and quantities used to teach the children number should be meaningful to them. Most seem to enjoy counting and comparing sets, such as deciding who among them has the most blocks. They should not be directly told to count, but events and activities should be arranged that encourage it; they may be told to bring ôenough cups for everybodyö, instead of being given the number to bring, so that they must compare the number of people to the number of cups needed; a common error of a child is to forget to include himself when counting; his reasoning, not his result, should be corrected; he could be asked if he remembered to count himself. It is of no benefit to correct an error that is unavoidable for a given stage, as the child will have no real understanding of what was done wrong, unless perhaps he discovers it for himself.

Peer interaction should also be encouraged. Often, through arguing or discussion, a correct answer will result that would not have been realized otherwise, as is true for most age groups.

Various situations can be set up that encourage mathematical reasoning. Children can be told to distribute materials among the class, requiring them to approximate numbers and differences. Dividing objects among a group fairly requires (or will improve) understanding of division. Keeping records, such as attendance, and voting to make class decisions can also be helpful.

Many common games require some understanding of mathematical principles. Scorekeeping in marbles and bowling requires comparison of quantities and simple operations. Musical chairs requires coordination between the set of children and the set of chairs; the importance is in the arrangement. Equalities are required in a guessing game where one child chooses a number between one and ten and another attempts to determine it, being told if his guesses are too high or too low. Board games, such as Candyland and Chutes & Ladders make use of cards, spinners, and die, all which possess numerical properties. Card games (War, Fives, Concentration) involve much comparison and calculation.

Beyond these general suggestions, a few curricula have been suggested, which, of course, are not endorsed by Piaget and not directly derived from his work. One suggests that little more than counting can be done in kindergarten; in first grade understanding of addition allows its first use; in second grade, two-digit addition and multiplication are possible; and in third grade, multiple-digit and multiple-term addition and multiplication.

V. Conclusions

The studies of Piaget reveal much of the advancement of human knowledge that often seems surprising, as is the nature of epistemology. A child's understanding of the universe seems rather random and unreasonable, but the advancement to operational logic occurs automatically through interaction with the environment, and can be accommodated in the classroom through application of Piaget's studies, primarily through directing the children's reasoning and providing real-world objects to stimulate understanding of relations that are necessary for mathematics.

On a more traditional epistemological note, the obviously flawed perceptions of young children must make us question the validity of our own; a child in the first stage would see it as reasonable; similarly, we see the third stage as reasonable; yet it is still somewhat perception-based, and many seemingly accurate views of the world have been proven wrong; the new ideas presented by quantum mechanics, such as the finite nature of space and nonexistence of trajectory motion, and those of physics, such as the curvature of space and relativity of time, are extremely unintuitive and show observable proof of the philosophical and epistemological concepts that have existed for millennia- that our perceptions and intuitions are unreliable. Fortunately, the field of mathematics has effectively minimized their importance. Little beyond the concept of conservation is assumed, which is the basis for most of human reasoning- yet its development was never considered before Piaget.

VI. Bibliography

1. Piaget, Jean. Child's Conception of Number. New York: W. W. Norton & Company, Inc, 1965.

2. Beard, Ruth M. An Outline of Piaget's Developmental Psychology for Students and Teachers. London: Routledge & Kegan Paul Ltd, 1969.

3. Kamii, Constance. Number in Preschool and Kindergarten: Educational Implications of Piaget's Theory. Washington, DC: National Association for the Education of Young Children, 1982.

4. Athey, Irene J. and Rubadeau, Duane O. Educational Implications of Piaget's Theory. Waltham: Ginn and Co, 1970.

5. Piaget, Jean. Science of Education and the Psychology of the Child. Paris: Editions Denoel, 1969.

6. Wadsworth, Barry J. Piaget's Theory of Cognitive Development. New York: Longman Inc, 1979.

7. Piaget, Jean. Epistemology and Psychology of Functions. Dordrecht: D. Reidel Publishing Company, 1977.

8. Sugarman, Susan. Piaget's Construction of the Child's Reality. Cambridge: Cambridge University Press, 1987.